The causal structure is basically a chaotic system, which means that NewtonIan style differential equations aren't much use, and big computerized models are. Ordinary weather forecasting uses big models, and I don't see why climate change, which is essentially very long term forecasting would different.

Climatological models and meteorological models are very different. If they weren't, then "we can't predict whether it will rain or not ten days from now" (which is mostly true) would be a slam-dunk argument against our ability to predict temperatures ten years from now. One underlying technical issue is that floating point arithmetic is only so precise, and this gives you an upper bound on the amount of precision you can expect from your simulation given the number of steps you run the model for. Thus climatological models have larger cells, larger step times, and so on, so that you can run the model for 50 model-years and still think the result that comes out might be reasonable.

(I also don't think it's right to say that Newtonian-style diffeqs aren't much use; the underlying update rules for the cells are diffeqs like that.)

Now suppose they each ask the question "what is the probability that, when doing what I did, one will come up with at most the number of tails I actually saw?"

That is throwing away data. The evidence that they each observed is the sequence of coin flip results, and the number of tails in that sequence is a partial summary of the data. The reason they get different answers is because that summary throws away more data for B than A. As you say, B already expected to get exactly one tail, so that summary tells him nothing new and he has no information to update on, while A can recover from this summary the number of heads and only loses information about the order (which cancels out anyways in the likelihood ratios between theories of independent coin flips). But if you calculate the probability that they each see that sequence you get the same answer for both, p(heads)^9999 * (1 - p(heads).

That is, the data gathering procedure is needed to interpret a partial summary of the data, but not the complete data.

Incidentally, Eliezer, I don't think you're right about the example at the beginning of the post. The two frequentist tests are asking distinct questions of the data, and there is not necessarily any inconsistency when we ask two different questions of the same data and get two different answers.

Suppose A and B are tossing coins. A and B both get the same string of results -- a whole bunch of heads (let's say 9999) followed by a single tail. But A got this by just deciding to flip a coin 10000 times, while B got it by flipping a coin until the first tail came up. Now suppose they each ask the question "what is the probability that, when doing what I did, one will come up with at most the number of tails I actually saw?"

In A's case the answer is of course very small; most strings of 10000 flips have many more than one tail. In B's case the answer is of course 1; B's method ensures that exactly one tail is seen, no matter what happens. The data was the same, but the questions were different, because of the "when doing what I did" clause (since A and B did different things). Frequentist tests are often like this -- they involve some sort of reasoning about hypothetical repetitions of the procedure, and if the procedure differs, the question differs.

If we wanted to restate this in Bayesian terms, we'd have to do so by taking into account that the interpreter knows what the method is, not just what the data is, and the distributions used by a Bayesian interpreter should take this into account. For instance, one would be a pretty dumb Bayesian if one's prior for B's method didn't say you'd get one tail with probability one. The observation that's causing us to update isn't "string of data," it's "string of data produced by a given physical process," where the process is different in the two cases.

(I apologize if this has all been mentioned before -- I didn't carefully read all the comments above.)

"Bayesianism's coherence and uniqueness proofs cut both ways. Just as any calculation that obeys Cox's coherency axioms (or any of the many reformulations and generalizations) must map onto probabilities, so too, anything that is not Bayesian must fail one of the coherency tests. This, in turn, opens you to punishments like Dutch-booking (accepting combinations of bets that are sure losses, or rejecting combinations of bets that are sure gains)."

I've never understood why I should be concerned about dynamic Dutch books (which are the justification for conditionalization, i.e., the Bayesian update). I can understand how static Dutch books are relevant to finding out the truth: I don't want my description of the truth to be inconsistent. But a dynamic Dutch book (in the gambling context) is a way that someone can exploit the combination of my belief at time (t) and my belief at time (t+1) to get something out of me, which doesn't seem like it should carry over to the context of trying to find out the truth. When I want to find the truth, I simply want to have the best possible belief in the present -- at time (t+1) -- so why should "money" I've "lost" at time (t) be relevant?

Perhaps I simply want to avoid getting screwed in life by falling into the equivalents of Dutch books in real, non-gambling-related situations. But if that's the argument, it should depend on how frequently such situations actually crop up -- the mere existence of a Dutch book shouldn't matter if life is never going to make me take it. Why should my entire notion of rationality be based on avoiding one particular -- perhaps rare -- type of misfortune? On the other hand, if the argument is that falling for dynamic Dutch books constitutes "irrationality" in some direct intuitive sense (the same way that falling for static Dutch books does), then I'm not getting it.